This question is just to satisfy my own curiosity. I play mainly guitar, and some piano. Both are chording instruments so I study chord structure and arpeggios as part of my education process. I found myself wondering whether students who study on instruments that cannot chord spend any time studying chord structure and whether arpeggio study is emphasized for them. I realize there is probably many different methods of instruction for non-chording instruments, I'm just curious as to whether chord structure and arpeggio studies are disregarded because of the inability to actually play chords?
Interesting question! Several of the people I play with who use 'non-chorded' instruments - trumpet, sax, clarinet and so on, have a far less interest in chords - and chord names in particular. It's most likely because although they can play broken chords - arpeggios - it's usually written down for them to read as they play. Therefore they don't see chords in the same light as piano or guitar players.
Those who play jazz seem to be more knowledgeable about chords. It's in their interests to know that when they see Dm7♭5 on a lead sheet, which notes it actually represents, therefore which notes are good to emphasise, or even arpeggiate.
It stands to reason that they will have learned chords in music school and may even play them on piano, but I think readers (i.e. non-improvisers) may not even be aware that what they play sequentially is a broken chord, whereas buskers and jazzers will realise knowing chords is a very useful tool in their armoury.
I play a lot of one arranger's pieces - he's a trumpet player - an excellent one too - and I know he's not too strong on recognising (and naming!) chords per se.
I think Tim's answer is very good but would add some to it.
In my guitar training I worked through the Lazarus clarinet method (with instructor guidance). Though I think it is a classical approach there are plenty of arpeggio exercises (most classical texts on any instrument have scale and arpeggio exercises).
I think your question can be interpreted in two ways, (1) do such players every study arpeggios or (2) do such players ever go into chord theory.
To the first interpretation I'd say everyone eventually studies arpeggios and the chords they represent, as arpeggios are a standard ingredient in constructing melodies. Clearly a horn player cannot play a 7 note chord in the sense of playing all notes simultaneously but the melodies and supporting voices in orchestral scores will be filled with arpeggios. A player who spends more time on sight reading and performing classical music may not delve into the full structure of music theory and harmony (at least not until they get into a serious music program). Classical lessons usually lean towards developing skills on the instrument and sight reading. On the other side of the spectrum (as Tim points out) there is modern performance, especially Jazz. Here the emphasis is on skills of course, but more is placed on understanding song structure, chords movement, and developing a good ear. This is because Jazz is improvisational by nature. That is not to say that classical musicians do not learn these things too, but the priorities are different. Jazz, at least they way I learned it in the 70s and 80s, is very much a sink or swim experience done live. While guitarists and pianists may be at an advantage physically in that their hands can feel their way around the changes horn players I know usually play through the changes of a new tune at least once quickly arpeggiating the chord changes (even throwing in cycle extensions) to get the chords in their ear. In this way the good ones have an advantage in that they need to "hear" it in their head and imo seem to develop good voice leading sometimes sooner than guitarists. Some guitarists can rely on the shapes of chords. There is nothing wrong with this but sometimes people forego the ear and rely on the hand.
A third point however is formal music education, as opposed to private lessons. Regardless of one's main instrument anyone who goes to music school will be exposed to (1) music theory, harmonization theory, and orchestration, and (2) piano lab. I know more than a hand full of professional musicians who play french horn, trombone, violin, etc, but earn a steady living playing piano thanks to music school. A horn player who primarily works in a classical orchestra will probably not rely on any music theory to perform but one who composes or arranges will need to know and understand chords (even if they cannot play them on the horn). An experienced arranger/composer will understand chords just as well as a guitarist (in some cases better since they are not prejudiced to the ones that are easiest to play).
So in some sense the answer to your question depends on the type of training and education the non-chordal instrumentalist has.